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# Interpretation of Benveniste’s experiments as a consequence of quantum-like interferences (for an experiment with an optimal interference term). If the sample labels are not blinded or blinded by a type-2 observer, then the cognitive state of A (described by the state vector | ψ A ) is able to

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Benveniste's experiments (also known as " memory of water " or " digital biology " experiments) remain unresolved. In some research areas, which have in common the description of cognition mechanisms and information processing, quantum-like statistical models have been proposed to address problems that were " paradoxical " in a classical frame. The...

## Contexts in source publication

**Context 1**

... this particular case, the observation of concordant pairs is always associated with label IN (i.e. IN is always associated with “ ↓ ”) and the observation of discordant pairs is always associated with label AC (i.e. AC is always associated with “ ↓ ”). Therefore, no signal is observed with commutable observables; only background is associated with both IN and AC labels. This shows that non-commutable observables are necessary not only for high rates of concordant pairs, but also for signal emergence. Note also that the signal must be one of the possible states of the system, even one with a low probability. In other words, the signal must be present in the background; thanks to entanglement, the emergence of the signal is made possible. In the proposed formalism, there is neither success nor failure of the experiments (Table 5). Simply, as in a single-particle interference experiment, we can decide to observe either “waves” or “particles” by modifying the setting of the experiment. In the two-slit experiment of Young, observing “waves” (interference pattern on the screen) is not considered as a success whereas observing “particles” (no interference pattern after which-path measurement) is not considered as a failure. In Benveniste’s experiments, the decision to observe “particles” (concordant pairs plus discordant pairs) or “waves” (only concordant pairs) is related to the design of the experiment (Figure 3). If the “cognitive state” of the experimenter is able to interfere with itself (as a single particle interferes with itself), then the probability of “success” is high. In case of blinding by a type-2 observer, quantum probabilities also apply since the respective cognitive states of the experimenter A and of the type-2 observer O are on the same branch of reality (as Wigner’s friend observing Schrödinger’s cat). Therefore, there is no formal difference for open-label vs. blinding by a type-2 observer. The same outcomes are obtained since the state vector that describes their cognitive states ...

**Context 2**

... In our previous article, we analyzed the experiments with the Langendorff system, and we concluded that they did not support the hypothesis of the “memory of water” (Beauvais 2012). We did not reach this conclusion because the known physical properties of water did not support memory in this liquid as argued by some authors (Teixeira 2007), but simply because a subset of results from Benveniste’s experiments themselves dismissed this hypothesis. In a fi rst step, we analyzed a set of experiments obtained by Benveniste’s team in the 1990s. We quanti fi ed the relationship between “expected” effects (i.e. labels of the tested samples) and apparatus outcomes, and we de fi ned the experimental conditions to observe signi fi cant correlations. We observed that the results were amazingly identical despite the various “stimuli” thought to induce a signal (high dilutions, direct “electromagnetic transfer” from a biological sample, “electromagnetic transfer” from a stored fi le, and transfer of the “biological activity” of homeopathic granules to water). Moreover, a diversity of electronic devices was used, particularly electric coils with various technical characteristics. In other words, the dynamic range of the “measure apparatus” used to evidence “informed water” seemed to be exceptionally large for the “input” but was nevertheless associated with a monotonous response for the “output.” What appeared to be the “cause” of the outcome was the “label” of the sample (“inactive” or “active”) and not the speci fi c physical process that had supposedly “informed” the water. We concluded that the results of these experiments were related to experimenter-dependent correlations, which did not support the initial “memory of water” hypothesis. Nevertheless, the fact that a signal emerged from background noise remained puzzling. Therefore, in a second step, we described Benveniste’s experiments according to the relational interpretation of quantum physics (Beauvais 2012). This interpretation allowed for the elaboration of a fi rst quantum approach of Benveniste’s experiments: The emergence of a signal from background noise was described by the entanglement of the experimenter with the observed system. Although our hypothesis did not de fi nitely dismiss the possibility of “memory of water,” the experimenter-dependent entanglement was an attractive alternative interpretation of Benveniste’s experiments. However, quick decoherence of any macroscopic system is an obstacle to the general acceptance of such an interpretation. In the next section, we propose a parallel between Benveniste’s experiments and classical interference experiments. This parallel allows for a description of a more complete formalism of Benveniste’s experiments. Single-particle quantum interference is one of the most important phenomena that illustrate the superposition principle and highlight the major difference between quantum and classical physics. The two-slit interferometer of Young can be used for one-particle interference experiments, but the Mach- Zehnder device has the advantage of ending only with two detectors (D1 and D2) and not with a screen (i.e. a great number of detectors) (Scarani & Suarez 1998). Figure 2 (upper drawing) depicts the Mach-Zehnder device. Light is emitted from a monochromatic light source: 50% of the light is transmitted by the beam splitter (BS1) in path T and 50% is re fl ected in path R. In BS2, the two beams are combined and 50% of the light is transmitted by the beam splitter in detector D1 and 50% in detector D2. If light is considered a wave, it can be calculated that waves from the two paths are constructive when they arrive in D1 and destructive in D2. Therefore, clicks after light detection are heard only in D1. This is indeed what experiment shows, and it is an argument for the wavy nature of light. On the contrary, if we consider light a collection of small balls (photons), they should randomly go into path T or R (with a probability of 0.5 for each path) and then in BS2 they go into D1 or D2 randomly (again with a probability of 0.5 for D1 or D2). As a consequence D1 should click in 50% of cases and D2 in 50% of cases. However, if photons are emitted one by one (by decreasing light intensity), the interference pattern persists (100% of clicks in D1). This is a quite counterintuitive result. Even more astonishingly, this unexpected (nonclassical) behavior disappears if the initial path (T or R) is detected by any means: Then either D1 or D2 clicks, each in 50% of cases (classical probabilities apply) (Figure 2, lower drawing). We made a parallel between Benveniste’s experiments and the one- particle interference experiment, which appeared to have isomorphic underlying mathematical structures. Indeed, according to the context of the experiment, either only concordant pairs (equivalent to detection in D1) or both concordant/discordant pairs (i.e. equivalent to random detection by D1 and D2) were obtained (Figure 3 and Table 3). The objective of our study is to describe the possible outcomes of the cognitive states of an experimenter in different contexts. Mathematically, a state is represented by a vector in a Hilbert space. Using the quantum formalism, the cognitive state of the experimenter is represented by the state vector | ψ A , which summarizes all the information on the quantum system. A key ingredient in the quantum formalism is the principle of superposition. According to this principle, the linear combination of any set of states is itself a possible state. Thus, if | A 1 and | A 2 are two possible states of the system, then | ψ A = λ 1 | A 1 + λ 2 | A 2 also is a possible state of A (with λ and λ real or complex numbers). This is due to the linearity of the 1 2 Schrödinger equation: Any linear combination of solutions to a particular equation will also be a solution to it. Therefore, a physical system exists in all its particular and theoretically possible states. When it is “measured,” only one state among the possible states is observed by the experimenter. The quantum formalism states that the probability to observe | A 1 is the square of the probability amplitude λ 1 associated with this state. An example of superposition that is directly observable is the interference pattern observed in the two-slit experiment. Interferences are the hallmark of superposed states and are the heart of quantum physics. Quantum interference is the consequence of non-commutable observables, as described in Figure 4. In a single-photon interference experiment, if one can (even in principle) distinguish the path each photon has taken, then interferences vanish and classical probabilities apply. In the setup depicted in Figure 2, the initial path cannot be distinguished in the upper drawing, and interferences occur; in the lower drawing, paths are distinguished by measurement, and consequently classical probabilities apply (without the interference term). The formalism of single-particle interference has been widely described and we propose to use it to describe Benveniste’s experiments (Table 3 and Figure 3). The distinction that we made between the type-1 (“outside”) observer and the type-2 (“inside”) observer is reminiscent of the thought experiment proposed by the physicist Eugene Wigner in the early 1960s and known as “Wigner’s Friend” (D’Espagnat 2005). In this thought experiment, Wigner’s friend performs a measurement on a quantum system in a superposed ...

**Context 3**

... noise was described by the entanglement of the experimenter with the observed system. Although our hypothesis did not de fi nitely dismiss the possibility of “memory of water,” the experimenter-dependent entanglement was an attractive alternative interpretation of Benveniste’s experiments. However, quick decoherence of any macroscopic system is an obstacle to the general acceptance of such an interpretation. In the next section, we propose a parallel between Benveniste’s experiments and classical interference experiments. This parallel allows for a description of a more complete formalism of Benveniste’s experiments. Single-particle quantum interference is one of the most important phenomena that illustrate the superposition principle and highlight the major difference between quantum and classical physics. The two-slit interferometer of Young can be used for one-particle interference experiments, but the Mach- Zehnder device has the advantage of ending only with two detectors (D1 and D2) and not with a screen (i.e. a great number of detectors) (Scarani & Suarez 1998). Figure 2 (upper drawing) depicts the Mach-Zehnder device. Light is emitted from a monochromatic light source: 50% of the light is transmitted by the beam splitter (BS1) in path T and 50% is re fl ected in path R. In BS2, the two beams are combined and 50% of the light is transmitted by the beam splitter in detector D1 and 50% in detector D2. If light is considered a wave, it can be calculated that waves from the two paths are constructive when they arrive in D1 and destructive in D2. Therefore, clicks after light detection are heard only in D1. This is indeed what experiment shows, and it is an argument for the wavy nature of light. On the contrary, if we consider light a collection of small balls (photons), they should randomly go into path T or R (with a probability of 0.5 for each path) and then in BS2 they go into D1 or D2 randomly (again with a probability of 0.5 for D1 or D2). As a consequence D1 should click in 50% of cases and D2 in 50% of cases. However, if photons are emitted one by one (by decreasing light intensity), the interference pattern persists (100% of clicks in D1). This is a quite counterintuitive result. Even more astonishingly, this unexpected (nonclassical) behavior disappears if the initial path (T or R) is detected by any means: Then either D1 or D2 clicks, each in 50% of cases (classical probabilities apply) (Figure 2, lower drawing). We made a parallel between Benveniste’s experiments and the one- particle interference experiment, which appeared to have isomorphic underlying mathematical structures. Indeed, according to the context of the experiment, either only concordant pairs (equivalent to detection in D1) or both concordant/discordant pairs (i.e. equivalent to random detection by D1 and D2) were obtained (Figure 3 and Table 3). The objective of our study is to describe the possible outcomes of the cognitive states of an experimenter in different contexts. Mathematically, a state is represented by a vector in a Hilbert space. Using the quantum formalism, the cognitive state of the experimenter is represented by the state vector | ψ A , which summarizes all the information on the quantum system. A key ingredient in the quantum formalism is the principle of superposition. According to this principle, the linear combination of any set of states is itself a possible state. Thus, if | A 1 and | A 2 are two possible states of the system, then | ψ A = λ 1 | A 1 + λ 2 | A 2 also is a possible state of A (with λ and λ real or complex numbers). This is due to the linearity of the 1 2 Schrödinger equation: Any linear combination of solutions to a particular equation will also be a solution to it. Therefore, a physical system exists in all its particular and theoretically possible states. When it is “measured,” only one state among the possible states is observed by the experimenter. The quantum formalism states that the probability to observe | A 1 is the square of the probability amplitude λ 1 associated with this state. An example of superposition that is directly observable is the interference pattern observed in the two-slit experiment. Interferences are the hallmark of superposed states and are the heart of quantum physics. Quantum interference is the consequence of non-commutable observables, as described in Figure 4. In a single-photon interference experiment, if one can (even in principle) distinguish the path each photon has taken, then interferences vanish and classical probabilities apply. In the setup depicted in Figure 2, the initial path cannot be distinguished in the upper drawing, and interferences occur; in the lower drawing, paths are distinguished by measurement, and consequently classical probabilities apply (without the interference term). The formalism of single-particle interference has been widely described and we propose to use it to describe Benveniste’s experiments (Table 3 and Figure 3). The distinction that we made between the type-1 (“outside”) observer and the type-2 (“inside”) observer is reminiscent of the thought experiment proposed by the physicist Eugene Wigner in the early 1960s and known as “Wigner’s Friend” (D’Espagnat 2005). In this thought experiment, Wigner’s friend performs a measurement on a quantum system in a superposed ...

## Citations

... However, further improvements of experimental conditions and devices did not prevent this unwanted phenomenon. 25,26 In 2013, I reanalyzed in depth a series of "digital biology" experiments with isolated rodent heart performed by Benveniste's team. 23 The main interest of this series of experiments was that both inside and outside supervisors operated on the same test samples. ...

The “memory of water” experiments suggested the existence of molecular-like effects without molecules. Although no convincing evidence of modifications of water – specific of biologically-active molecules – has been reported up to now, consistent changes of biological systems were nevertheless recorded. We propose an alternate explanation based on classical conditioning of the experimenter. Using a probabilistic model, we describe not only the biological system, but also the experimenter engaged in an elementary dose-response experiment. We assume that during conventional experiments involving genuine biologically-active molecules, the experimenter is involuntarily conditioned to expect a pattern, namely a relationship between descriptions (or “labels”) of experimental conditions and corresponding biological system states. The model predicts that the conditioned experimenter could continue to record the learned pattern even in the absence of the initial cause, namely the biologically-active molecules. The phenomenon is self-sustained because the observation of the expected pattern reinforces the initial conditioning. A necessary requirement is the use of a system submitted to random fluctuations with autocorrelated successive states (no forced return to the initial position). The relationship recorded by the conditioned experimenter is, however, not causal in this model because blind experiments with an “outside” supervisor lead to a loss of correlations (i.e., system states randomly associated to “labels”). In conclusion, this psychophysical model allows explaining the results of “memory of water” experiments without referring to water or another local cause. It could be extended to other scientific fields in biology, medicine and psychology when suspecting an experimenter effect.

... The cumulative results from 886 experiments, both blinded and open, conducted by Benveniste's group over almost a decade, have been presented and analysed by Beauvais. 6,16,33 These experiments, pertaining to the field named by Benveniste as 'digital biology', concern the transmission of the activity of a potentised solution of a pharmacological agent to pure water through an electromagnetic signal and then testing if this transmission did in fact occur, by means of an isolated heart model, using the 'Langendorff Homeopathy Vol. 107 No. 3/2018 apparatus'. ...

We discuss questions related to the ‘Benveniste Affair’, its consequences and broader issues in an attempt to understand homeopathy. Specifically, we address the following points:
1. The relationship between the experiments conducted by Benveniste, Montagnier, their collaborators and groups that independently tested their results, to ‘traditional’ homeopathy.
2. Possible non-local components such as ‘generalised entanglement’ as the basis of the homeopathic phenomenon and experimental evidence for them.
3. The capability of highly diluted homeopathic remedies to provoke tangible biological changes in whole organisms and cellular experimental systems.
4. Aspects of the similia principle related to the above.
5. Suggestions that can lead to experimental verifications of the non-local hypothesis in homeopathy.

... 25 This modeling was an adaptation of a previous model aimed to describe Benveniste's in vitro experiments. 26 Most physicians and biologists are admittedly unenthusiastic to read articles with mathematical reasoning. The quantum formalism conveys counterintuitive notions that are described with unfamiliar mathematical tools (Hilbert's space, state vectors, non commutative observables, etc). ...

In previous articles, a description of 'unconventional' experiments (e.g.
in vitro or clinical studies based on high dilutions, 'memory of water' or
homeopathy) using quantum-like probability was proposed. Because the mathematical
formulations of quantum logic are frequently an obstacle for physicians and
biologists, a modified modeling that rests on classical probability is described
in the present article. This modeling is inspired from a relational
interpretation of quantum physics that applies not only to microscopic objects,
but also to macroscopic structures, including experimental devices and observers.
In this framework, any outcome of an experiment is not an absolute property of
the observed system as usually considered but is expressed relatively to an
observer. A team of interacting observers is thus described from an external view
point based on two principles: the outcomes of experiments are expressed
relatively to each observer and the observers agree on outcomes when they
interact with each other. If probability fluctuations are also taken into
account, correlations between 'expected' and observed outcomes emerge. Moreover,
quantum-like correlations are predicted in experiments with local blind design
but not with centralized blind design. No assumption on 'memory' or other
physical modification of water is necessary in the present description although
such hypotheses cannot be formally discarded. In conclusion, a simple modeling of
'unconventional' experiments based on classical probability is now available and
its predictions can be tested. The underlying concepts are sufficiently intuitive
to be spread into the homeopathy community and beyond. It is hoped that this
modeling will encourage new studies with optimized designs for in vitro
experiments and clinical trials.

... Das war der direkte Versuch, die Hypothese von Francis Beauvais umzusetzen [34][35][36]. Francis Beauvais war ein Mitarbeiter und Kollege von Jacques Benveniste. In seinen Publikationen analysiert er das Scheitern des Benveniste'schen Programms, das «Gedächtnis des Wassers» oder später der «digital biology» zu beweisen. ...

... Details of these public demonstrations organized from 1992 to 1997 with the rodent isolated heart model have been given elsewhere [7,21,23] and one of them has been thoroughly analyzed in a recent article [22]. So what did not work in these demonstrations? ...

The case of the “memory of water” was an outstanding scientific controversy of the end of the twentieth century which has not been satisfactorily resolved. Although an experimenter effect has been proposed to explain Benveniste’s experiments, no evidence or convincing explanation supporting this assumption have been reported. One of the unexplained characteristics of these experiments was the different outcomes according to the conditions of blinding. In this article, an original probabilistic modeling of these experiments is described that rests on a limited set of hypotheses and takes into account measurement fluctuations. All characteristics of these disputed results can be described, including their “paradoxical” aspects; no hypothesis on changes of water structure is necessary. The results of the disputed Benveniste’s experiments appear to be a misinterpreted epiphenomenon of a more general phenomenon. Therefore, this reappraisal of Benveniste’s experiments suggests that these results deserved attention even though the hypothesis of “memory of water” was not supported. The experimenter effect remains largely unexplored in biosciences and this modeling could give a theoretical framework for some improbable, unexplained or poorly reproducible results.

... The protocol of these experimental demonstrations was designed as ''proof of concept'' to hopefully give a definitive confirmation on the reality of ''electronic transmission'' or ''digital biology''. Details on these demonstrations have been given elsewhere (Beauvais 2007(Beauvais , 2012(Beauvais , 2013a; the results of one public demonstration has been thoroughly analyzed in a recent article (Beauvais 2013b). ...

... Such a series of experiments with both Bob and Eve who assessed the results has been described in detail in a previous article (Beauvais 2013b). Description of Benveniste's experiments in different experimental conditions (with or without Eve's assessment) have been reported elsewhere (Beauvais 2012(Beauvais , 2013a. ...

... I proposed in recent articles to describe Benveniste's experiments in a probabilistic quantum-like modeling (Beauvais 2012(Beauvais , 2013a(Beauvais , b, 2014. The price to pay was to abandon the notions of ''memory of water'', ''digital biology'', and so on. ...

Benveniste’s experiments were at the origin of a scientific controversy that has never been satisfactorily resolved. Hypotheses based on modifications of water structure that were proposed to explain these experiments (“memory of water”) were generally considered as quite improbable. In the present paper, we show that Benveniste’s experiments violated the law of total probability, one of the pillars of classical probability theory. Although this could suggest that quantum logic was at work, the decoherence process is however at first sight an obstacle to describe this macroscopic experimental situation. Based on the principles of a personalist view of probability (quantum Bayesianism or QBism), a modeling could nevertheless be built that fitted the outcomes reported in Benveniste’s experiments. Indeed, in QBism, there is no split between microscopic and macroscopic, but between the world where an agent lives and his internal experience of that world. The outcome of an experiment is thus displaced from the object to its perception by an agent. By taking into account both the personalist view of probability and measurement fluctuations, all characteristics of Benveniste’s experiments could be described in a simple modeling: change of the biological system from resting state to “activated” state, concordance of “expected” and observed outcomes and apparent “jumping” of “biological activities” from sample to sample. No hypothesis on change of water structure was necessary. In conclusion, a modeling of Benveniste’s experiments based on a personalist view of probability offers for the first time a logical framework for these experiments that have remained controversial and paradoxical till date.

... Overall, this formalism fits the corpus of the experimental data gained by Benveniste's team over the years [2,13,15,23]. Moreover, in this modeling, no physico-chemical explanation such as "memory of water" is necessary. ...

Benveniste’s experiments have been the subject of an international scientific controversy (known as the case of the “memory of water”). We recently proposed to describe these results in a modeling in which the outcome of an experiment is considered personal property (named cognitive state) of the observer and not an objective property of the observed system. As a consequence, the correlations between “expected” results and observed results in Benveniste’s experiments could be considered the consequence of quantum-like interferences of the possible cognitive states of the experimenters/observers.
In the present paper, we evidence that small random fluctuations from the environment together with intersubjective agreement force the “expected” results and the observed results experienced by the observers into a noncommuting relationship. The modeling also suggests that experimental systems with enough compliance (e.g., biological systems) are more suitable to evidence quantum-like correlations. No hypothesis related to “memory of water” or other elusive modifications of water structure is necessary.
In conclusion, a quantum-like interpretation of Benveniste’s experiments offers a logical framework for these experiments that have remained paradoxical to now. This quantum-like modeling could be adapted to other areas of research for which there are issues of reproducibility of results by other research teams and/or suspicion of nontrivial experimenter effect.

... Quantum-like correlations of the "cognitive states" of the experimenter To solve the dilemma described in the previous section, we proposed in a series of articles to describe these experiments using notions from quantum logic (Beauvais, 2012;2013a;Beauvais, 2013c;. In the model that we described, there is no need of postulating "memory of water". ...

... In our previous articles (Beauvais, 2013a; b; c), we showed that the paradoxes of Benveniste's experiments disappeared if the possible "cognitive states" of the experimenter were described according to some principles from quantum physics (superposition and probability interferences). We obtained equations that correctly described the characteristics of Benveniste's experiments, namely emergence of signal from background noise, disturbance of blind experiments (type 1 vs. type 2) and difficulties for other teams to reproduce the experiments. ...

In previous articles, we proposed to describe the results of Benveniste’s experiments using a theoretical framework based on quantum logic. This formalism described all characteristics of these controversial experiments and no paradox persisted. This interpretation supposed to abandon an explanation based on a classical local causality such as the “memory of water hypothesis. In the present article, we describe with the same formalism the cognitive states of different experimenters who interact together. In this quantum-like model, the correlations observed in Benveniste’s experiments appear to be the consequence of the intersubjective agreement of the experimenters.

... In a second article, we showed that Benveniste's experiments and quantum interference experiments of single particles had the same logical structure. This parallel allowed elaborating a more complete formalism of Benveniste's experiments and we proposed to see Benveniste's experiments as the result of quantum-like probability interferences of cognitive states (Beauvais 2013). ...

... Biological systems will not be detailed and will be considered as black boxes with inputs (sample labels) and outputs (biological outcomes); only the logical aspects and the underlying mathematical structures of these experiments will be taken into account. Some of the ideas presented here have been previously published, but the present article offers a synthesis and takes a closer look at specific issues raised by the quantum-like formalism that were not addressed before (Beauvais 2012(Beauvais , 2013. ...

... In one case (interference pattern), light behaves as a wave and in the other case (no interference pattern), it behaves as a collection of particles. In Benveniste's results, the experimental context also appeared to play an important role (blinding by a type-1 observer vs. a type-2 observer) (Beauvais 2007(Beauvais , 2008(Beauvais , 2012(Beauvais , 2013. According to the blinding conditions, different results were obtained that were considered as ''successes'' or ''failures'' (Table 3). ...

The “memory of water” was a major international controversy that remains unresolved. Taken seriously or not, this hypothesis leads to logical contradictions in both cases. Indeed, if this hypothesis is held as wrong, then we have to explain how a physiological signal emerged from the background and we have to elucidate a bulk of coherent results. If this hypothesis is held as true, we must explain why these experiments were difficult to reproduce by other teams and why some blind experiments were so disturbing for the expected outcomes. In this article, a third way is proposed by modeling these experiments in a quantum-like probabilistic model. It is interesting to note that this model does not need the hypothesis of the “memory of water” and, nevertheless, all the features of Benveniste’s experiments are taken into account (emergence of a signal from the background, difficulties faced by other teams in terms of reproducibility, disturbances during blind experiments, and apparent “jumps of activity” between samples). In conclusion, it is proposed that the cognitive states of the experimenter exhibited quantum-like properties during Benveniste’s experiments.

... The present article should not induce any hypertensive response since I will describe a series of Benveniste's experiments without reference to modification of water structure whatsoever. Indeed, I proposed recently to model these controversial experiments with some notions inspired from the generalized probability theory that is the core of quantum physics (Beauvais, 2012;2013). Strictly speaking, the possibility of "memory of water" was not definitely dismissed; it is always difficult to prove that something does not exist. ...

... The formulas of PII (ADP) and PI (ADP) are similarly calculated: PII (ADP) = |b cos θ -a sin θ| 2 PI (ADP) = b 2 cos 2 θ + a sin 2 θ with PI (ADP|AIN) = sin 2 θ and PI (ADP|AAC) = cos 2 θ In a previous paper, this model allowed describing Benveniste's experiments without any reference to "memory of water", "electronic transmission", "digital biology" or any other "local" explanation (Beauvais, 2013). Just supposing superposed states and noncommutable observables, the quantum-like model described the main characteristics of Benveniste's experiments: emergence of signal from background, different outcomes according to type-1 or type-2 blinding and apparent "jumps of activity" between samples. ...

... Thanks to entanglement, the probability of signal increases. In a previous article, we proposed that the relationship between different cognitive states (AIN with A↓ and AAC with A↑), which are summarized in θ value, results of associative processes related to cognition mechanisms (Beauvais, 2013). ) |a cos θ + b sin θ| 2 a 2 cos 2 θ + b 2 sin 2 θ a 2 Probability of discordant pairs: P(A DP ) |b cos θ -a sin θ| 2 b 2 cos 2 θ + a 2 sin 2 θ b 2 ...

Objectives: “Memory of water” experiments (also known as Benveniste’s experiments) were the source of a famous controversy in the contemporary history of sciences. We recently proposed a formal framework devoid of any reference to “memory of water” to describe these disputed experiments. In this framework, the results of Benveniste’s experiments are seen as the consequence of quantum-like interferences of cognitive states. Design:
In the present article, we describe retrospectively a series of experiments in physiology (Langendorff preparation) performed in 1993 by Benveniste’s team for a public demonstration. These experiments aimed at demonstrating “electronic transmission of molecular information” from protein solution (ovalbumin) to naïve water. The experiments were closely controlled and blinded by participants not belonging to Benveniste’s team. Results: The number of samples associated with signal (change of coronary flow of isolated rodent heart) was as expected; this was an essential result since, according to mainstream science, no effect at all was supposed to occur. However, besides coherent correlations, some results were paradoxical and remained incomprehensible in a classical
framework. However, using a quantum-like model, the probabilities of the different outcomes could be calculated according to the different experimental contexts. Conclusion: In this reassessment of an historical series of memory of water” experiments, quantum-like probabilities allowed modeling these controversial experiments that remained unexplained in a classical frame and no logical paradox persisted. All the features of Benveniste’s experiments were taken into account with this model, which did not involve the hypothesis of “memory of water” or any other “local” explanation.